Our Metropolis MC scheme is preformed based on Eq. 19. At each step,
\(H_{p}\)  are calculated to determine the rejection
and acceptance. \(det\Lambda \)  is calculated by a certain algorithm
with the computational cost scaled by \(N^{3}\) .[1]} This kind of
numerical technique enables us to perform the fermionic simulations
with reasonable computational time. There are two basic types of
moves in current simulations: (1) Displacement move, where all the
coordinates for a single particle are displaced uniformly; (2)
Standard bisection moves.[2]}, [3]} The MC
procedure used in this work is wildly used by others. One MC step is
defined as one application of each procedure. Ten million MC steps
of calculation were carried out for each temperature. For a few
cases, 100 million MC steps are made to check the ergodic problem.
The results agree with the short runs within the error bars. To
further check the ergodic problem, the simulations are carried out
by a few random generated starting configurations. All simulations
converge to the same results. The energy is calculated based on the
thermodynamic estimator.[2]} The energies are well
converged at M/\(\beta \) =20, 22 and 5 for case 1, 2 and 3
respectively.
